been proved that the value of the function at all points of its region
of existence can be obtained from its value, supposed given by a
series in one original circle, by a succession of such processes of
analytical continuation.
§ 7. Monogenic Functions.—This suggests an entirely different way of formulating the fundamental parts of the theory of functions of a complex variable, which appears to be preferable to that so far followed here.
Starting with a convergent power series, say in powers of z, this series can be arranged in powers of z − z0, about any point z0 interior to its circle of convergence, and the new series converges certainly for |z − z0| < r − |z0|, if r be the original radius of convergence. If for every position of z0 this is the greatest radius of convergence of the derived series, then the original series represents a function existing only within its circle of convergence. If for some position of z0 the derived series converges for |z − z0| < r − |z0| + D, then it can be shown that for points z, interior to the original circle, lying in the annulus r − |z0| < |z − z0| < r − |z0| + D, the value represented by the derived series agrees with that represented by the original series. If for another point z1 interior to the original circle the derived series converges for |z − z1| < r − |z1| + E, and the two circles |z − z0| = r − |z0| + D, |z − z1| = r − |z1| + E have interior points common, lying beyond |z| = r, then it can be shown that the values represented by these series at these common points agree. Either series then can be used to furnish an analytical continuation of the function as originally defined. Continuing this process of continuation as far as possible, we arrive at the conception of the function as defined by an aggregate of power series of which every one has points of convergence common with some one or more others; the whole aggregate of points of the plane which can be so reached constitutes the region of existence of the function; the limiting points of this region are the points in whose neighbourhood the derived series have radii of convergence diminishing indefinitely to zero; these are the singular points. The circle of convergence of any of the series has at least one such singular point upon its circumference. So regarded the function is called a monogenic function, the epithet having reference to the single origin, by one power series, of the expressions representing the function; it is also sometimes called a monogenic analytical function, or simply an analytical function; all that is necessary to define it is the value of the function and of all its differential coefficients, at some one point of the plane; in the method previously followed here it was necessary to suppose the function differentiable at every point of its region of existence. The theory of the integration of a monogenic function, and Cauchy’s theorem, that ∫ƒ(z)dz = 0 over a closed path, are at once deducible from the corresponding results applied to a single power series for the interior of its circle of convergence. There is another advantage belonging to the theory of monogenic functions: the theory as originally given here applies in the first instance only to single valued functions; a monogenic function is by no means necessarily single valued—it may quite well happen that starting from a particular power series, converging over a certain circle, and applying the process of analytical continuation over a closed path back to an interior point of this circle, the value obtained does not agree with the initial value. The notion of basing the theory of functions on the theory of power series is, after Newton, largely due to Lagrange, who has some interesting remarks in this regard at the beginning of his Théorie des fonctions analytiques. He applies the idea, however, primarily to functions of a real variable for which the expression by power series is only of very limited validity; for functions of a complex variable probably the systematization of the theory owes most to Weierstrass, whose use of the word monogenic is that adopted above. In what follows we generally suppose this point of view to be regarded as fundamental.
§ 8. Some Elementary Properties of Single Valued Functions.—A pole is a singular point of the function ƒ(z) which is not a singularity of the function 1/ƒ(z); this latter function is therefore, by the definition, capable of representation about this point, z0, by a series [ƒ(z)]−1 = Σan(z − z0)n. If herein a0 is not zero we can hence derive a representation for ƒ(z) as a power series about z0, contrary to the hypothesis that z0 is a singular point for this function. Hence a0 = 0; suppose also a1 = 0, a2 = 0, ... am−1 = 0, but am ± 0. Then [ƒ(z)]−1 = (z − z0)m[am + am+1 (z − z0) + ...], and hence (z − z0)mƒ(z) = am−1 + Σbn (z − z0)n, namely, the expression of ƒ(z) about z = z0 contains a finite number of negative powers of z − z0 and a (finite or) infinite number of positive powers. Thus a pole is always an isolated singularity.
The integral ∫ƒ(z)dz taken by a closed circuit about the pole not containing any other singularity is at once seen to be 2πiA1, where A1 is the coefficient of (z − z0)−1 in the expansion of ƒ(z) at the pole; this coefficient has therefore a certain uniqueness, and it is called the residue of ƒ(z) at the pole. Considering a region in which there are no other singularities than poles, all these being interior points, the integral 12πi ∫ ƒ(z)dz round the boundary of this region is equal to the sum of the residues at the included poles, a very important result. Any singular point of a function which is not a pole is called an essential singularity; if it be isolated the function is capable, in the neighbourhood of this point, of approaching arbitrarily near to any assigned value. For, the point being isolated, the function can be represented, in its neighbourhood, as we have proved, by a series Σ ∞−∞ an(z − z0)n; it thus cannot remain finite in the immediate neighbourhood of the point. The point is necessarily an isolated essential singularity also of the function {ƒ(z) − A}−1 for if this were expressible by a power series about the point, so would also the function ƒ(z) be; as {ƒ(z) − A}− 1 approaches infinity, so does ƒ(z) approach the arbitrary value A. Similar remarks apply to the point z = ∞, the function being regarded as a function of ζ = z−1. In the neighbourhood of an essential singularity, which is a limiting point also of poles, the function clearly becomes infinite. For an essential singularity which is not isolated the same result does not necessarily hold.
A single valued function is said to be an integral function when it has no singular points except z = ∞. Such is, for instance, an integral polynomial, which has z = ∞ for a pole, and the functions exp (z) which has z = ∞ as an essential singularity. A function which has no singular points for finite values of z other than poles is called a meromorphic function. If it also have a pole at z = ∞ it is a rational function; for then, if a1, ... as be its finite poles, of orders m1; m2, ... ms, the product (z − a1)m1 ... (z − as) msƒ(z) is an integral function with a pole at infinity, capable therefore, for large values of z, of an expression (z−1)−m Σ r=0 ar(z−1)r; thus (z − a1)m1 ... (z − as)msƒ(z) is capable of a form Σ r=0 brzr, but z−m Σ r=0 brzr remains finite for z = ∞. Therefore br+1 = br+2 = ... = 0, andƒ(z) is a rational function.
If for a single valued function F(z) every singular point in the finite part of the plane is isolated there can only be a finite number of these in any finite part of the plane, and they can be taken to be a1, a2, a3, ... with |a1| ⋜ |a2| ⋜ |a3| ... and limit |an| = ∞. About as the function is expressible as Σ ∞−∞ An(z − as)n; let ƒs(z) = Σ 1−∞ An(z − as)n be the sum of the negative powers in this expansion. Assuming z = 0 not to be a singular point, let ƒs(z) be expanded in powers of z, in the form Σ n=0 Cnzn, and μs be chosen so that Fs(z) = ƒs(z) − Σ μs−11 Cnzn = Σ ∞μs Cnzn is, for |z| < rs < |as|, less in absolute value than the general term εs of a fore-agreed convergent series of real positive terms. Then the series φ(z) = Σ ∞s=1 Fs(z) converges uniformly in any finite region of the plane, other than at the points as, and is expressible about any point by a power series, and near as, φ(z) − fs(z) is expressible by a power series in z − as. Thus F(z) − φ(z) is an integral function. In particular when all the finite singularities of F(z) are poles, F(z) is hereby expressed as the sum of an integral function and a series of rational functions. The condition |Fs(z)| < εs is imposed only to render the series ΣFs(z) uniformly convergent; this condition may in particular cases be satisfied by a series Σ Gs(z) where Gs(z) = ƒs(z) − Σ νs−11 Cnzn and νs < μs. An example of the theorem is the function π cot πz − z− 1 for which, taking at first only half the poles, ƒs(z) = 1/(z − s); in this case the series Σ Fs(z) where Fs(z) = (z − s)−1 + s−1 is uniformly convergent; thus π cot πz − z−1 − Σ ∞−∞ [(z − s)−1 + s−1], where s = 0 is excluded from the summation, is an integral function. It can be proved that this integral function vanishes.
Considering an integral function ƒ(z), if there be no finite positions of z for which this function vanishes, the function λ[ƒ(z)] is at once seen to be an integral function, φ(z), or ƒ(z) = exp[φ(z)]; if however great R may be there be only a finite number of values of z for which ƒ(z) vanishes, say z = a1, ... am, then it is at once seen that ƒ(z) = exp [φ(z)]. (z − a1)h1...(z − am)hm, where φ(z) is an integral function, and h1, ... hm are positive integers. If, however, ƒ(z) vanish for z = a1, a2 ... where |a1| ⋜ |a2| ⋜ ... and limit |an| = ∞, and if for simplicity we assume that z − 0 is not a zero and all the zeros a1, a2, ... are of the first order, we find, by applying the preceding theorem to the function [1 / ƒ(z)] [dƒ(z) / dz], that ƒ(z) = exp [φ(z)] Π ∞n=1 {(1 − z/an) exp φn(z)}, where φ(z) is an integral function, and φn(z) is an integral polynomial of the form φn(z) = z/an + z2/2an2 + ... + zs/sans. The number s may be the same for all values of n, or it may increase indefinitely with n; it is sufficient in any case to take s = n. In particular for the function
